'parroting' ultimately comes from the noun 'parrot', which in English was borrowed from Middle French 'perroquet'.

'perroquet' (Middle French) became English 'parrot' in Early Modern English; the verb 'to parrot' was later formed from the noun, and 'parroting' developed by adding the English gerund/nominalizing suffix '-ing'.

Originally the term referred to the bird ('parrot'); over time the bird's ability to imitate human speech led to the verb sense 'to imitate or repeat' and the noun 'parroting' meaning mechanical repetition or mimicry.